Karin Baur
Ruhr University Bochum

Frieze patterns, surfaces and representation theory

Abstract. Triangulations of polygons correspond to Coxeter's frieze patterns. These are finite frieze patterns and they are linked to cluster categories in type A. We explain how infinite frieze patterns arise from triangulations of surfaces and discuss their growth behaviour. In particular, we show that in affine types, this growth is well behaved and can be viewed as an invariant of the associated cluster category.

Wen Chang
Shaanxi Normal University

Tilting-completion for gentle algebras

Abstract. It is proved that any almost tilting module over a gentle algebra is partial tilting, that is, it can be completed to a tilting module. Furthermore, it has at most 2n complements, which partially confirms a conjecture of Happel for the case of gentle algebras. At the same time, for any n ≥ 3 and 1 ≤ m ≤ n–2, there always exists a connected gentle algebra with rank n and a pre-tilting module over it with rank m which is not partial tilting. The tool we use is the surface model associated with the module category of a gentle algebra. The main technique is doing inductions by cutting the surface, which is expected to be useful elsewhere.

Steffen Koenig
University of Stuttgart

Homological stability for diagram algebras 

Abstract. This talk is about homological stability in the sense of Nakaoka stability. Given a sequence of algebraic objects (indexed by n), for instance symmetric groups, related by natural maps, one considers graded homological invariants, for instance homology, aka Tor(k,k) groups. The sequence is said to satisfy homological stability, if the induced maps between degree i homology are isomorphisms for n large enough with respect to i. In 1960, Nakaoka proved homological stability for symmetric groups.
In 2021, Boyd, Hepworth and Patzt used Nakaoka's result to establish homological stability for Brauer algebras. Actually they proved a strong version when a certain parameter is invertible, and a weaker version – needing different methods – otherwise. Similar results have been obtained for some other diagram algebras.
In joint work in progress with Anne Henke and Geetha Thangavelu, algebraic structures behind such results are explained that yield the strong version of homological stability for Brauer algebras and other diagram algebras, and that explain why the weaker version needs to be weaker.

Chris Parker
Bielefeld University

New developments on existence and classification results for t-structures

Abstract. I will survey recent developments for (non-)existence, classification, and obstruction related results regarding t-structures on triangulated categories coming from algebraic geometry and representation theory. In particular, relating to Neeman's solution of a bold conjecture of Antieau, Gepner, and Heller regarding the existence of bounded t-structures on triangulated categories coming from finite dimensional Noetherian schemes, and a recent generalisation of this theorem to triangulated categories with finite finitistic dimension. This generalised version of Neeman's theorem has many applications in new contexts, for example: associative rings, connective E₁-rings, non-positive DG rings, algebraic stacks, and even triangulated categories without models. I will also talk about classification results for such t-structures, generalizing some recent work of Takahashi on finite dimensional Cohen–Macaulay excellent rings, and of Smith for finite dimensional Noetherian rings, to non-affine settings. These results are joint work with many people which spans a few articles, namely Rudradip Biswas, Hongxing Chen, Alex Clark, Pat Lank, Kabeer Manali Rahul, and Junhua Zheng.

Kyungmin Rho
Paderborn University

Geometric Fukaya category of surfaces and its applications to representation theory

Abstract. We broadly sketch homological mirror symmetry of the geometric Fukaya category of surfaces (A-side) with two grading structures: Z-grading and Z₂-grading. In the Z-graded case, it is related to the derived category of gentle algebras, and to the derived category of coherent sheaves on non-commutative singular curves (B-side). In the Z₂-graded case, it is related to the category of matrix factorizations of Landau–Ginzburg models, and to the category of Cohen–Macaulay sheaves on normal crossing surface singularities (B-side). We discuss representation of objects in each category and their connections with some concrete examples.